Abstract:The aim of this paper is to construct robust and accurate hybrid FVS/FDS type of schemes for a standard four-equation isentropic compressible two-fluid model governing 1-dimensional flow of a gas ($\text{g}$) and liquid ($\text{l}$) mixture. The starting point for our investigations is a Roe scheme and a hybrid FVS/FDS scheme. The latter is an AUSMD type of scheme obtained through a natural and rather straightforward extension of the corresponding scheme for the Euler equations (single-phase model) as described by Wada and Liou (1997, SIAM~J.~Sci.~Comput.~{18}, 633--657). The main advantage of such hybrid FVS/FDS schemes is that they neither require the use of Riemann solvers nor the computation of nonlinear flux Jacobians. However, we observe that the two-phase AUSMD scheme is prone to introducing oscillations and overshoots around discontinuities. Based on the belief that this deficiency is due to the loose coupling between mass and momentum equations in the discretization of the two-phase model, we propose a method for improving the approximation properties of hybrid FVS/FDS schemes by enforcing a tighter coupling between the various equations.The method, which is denoted as a "Mixture Flux" (MF) method, is composed of two main ingredients. First, we make use of an additional pressure evolution equation which is derived from the equation describing the conservation of the total mass. This provides us with information how to construct an appropriate numerical flux for the discretization of the pressure term of the momentum equations. Second, we introduce a consistent decomposition of the numerical mass fluxes into two components; one flux component $F^{\text{D}}$ associated with the fast-moving pressure waves and another component $F^{\text{A}}$ associated with the slow-moving volume fraction waves. The $F^{\text{D}}$-component is designed by using information from the momentum equations and is crucial for ensuring non-oscillatory behavior around the slow-moving volume fraction waves, whereas the $F^{\text{A}}$-component is responsible for the accuracy of these waves.

Particularly, by associating the flux $F^{\text{A}}$ with the AUSMD mass flux we demonstrate through numerical experiments that the resulting MF-AUSMD scheme possesses accuracy and stability properties on the same level as the Roe scheme while allowing for highly improved computational efficiency. In addition, by using a slight modification of MF-AUSMD we can also simulate flow cases which locally involve transition from two-phase to single-phase.

The MF-method represents a general strategy for refining hybrid FVS/FDS schemes for two-phase flow models.

**Paper:**- Available as PDF (856 Kbytes).
**Author(s):**- Steinar Evje, <Steinar.Evje@rf.no>
- Tore Flåtten, <toref@ifi.uio.no>
**Publishing information:****Comments:****Submitted by:**- <toref@ifi.uio.no> December 9 2003.

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